1. Boolean Algebra, Bitwise Operations
Tutorial Two
Boolean Algebra, Bitwise Operations
CompSci Semester Two 2017

2. Boolean Values Two types: Generally given the familiar values 1 and 0
True
False
Generally given the familiar values 1 and 0
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3. Boolean Operators AND OR NOT 𝐴 𝑇 Symbol: ∙ Conjunction Symbol: +
T ∙ T = T, T ∙ F = F, 1 ∙ 1 = 1 OR
Symbol: +
Disjunction
T + T = T, T + F = T, = 0
NOT
Symbol: line over the operator
Negation
= F, = 1 𝐴 𝑇
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4. Truth Tables
AND A B
A ∙ B 1
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5. Truth Tables OR A B
A + B 1
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6. Truth Tables
NOT A 1 𝐴
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7. Bitwise Operations Work on bit patterns, individual bits
Manipulate values for comparisons/Calculations
Bit Masking
Very Fast (Directly supported by CPU)
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8. Bitwise Operators XOR AND OR NOT Symbol: ^ Exclusive OR Symbol: &
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9. Bitwise operations XOR 1000 ^ 1011 = 0011 0111 ^ 1010 = 1101
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10. Bitwise operations AND 1000 & 1011 = 1000 0111 & 1000 = 0000
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11. Bitwise operations OR 1000 | 1011 = 1011 0111 | 1000 = 1111
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12. Bitwise operations NOT ~1010 = 0101 ~0111 = 1000
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13. ADDITION
Take the two binary numbers 01 (1) and 10 (2) To add these two numbers, first align them vertically Then, as with adding two decimal numbers, start from the right, and go left Note: The above is not in two’s compliment.
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14. Addition Cont.
What happens if we have two 1’s in the same column? Like in decimal addition, you place the remainder of the addition in that column, and then the carry into the column one to its left (1)00 What’s with the brackets?
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15. Subtraction
Subtraction is a subset of addition, in that adding a numbers negative is the same as subtracting that number 10 – 5 = (-5) = 5 The same is true for binary arithmetic. This is something important to remember for later when working with the LC-3, as it doesn’t have a subtraction instruction.
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16. Exercise 1
Calculate the result of the following 2’s Compliment arithmetic
1111 −0001
0000 −1010
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17. Solution 1 B
“ ”
left pad any numbers so both have the same number of bits
0111 =
Now align them vertically, and add the
000111
(carry 1)
(Carry 1) …
(1)
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18. Exercise 1 Solutions 10 + 1111 (-2 + -1) 0111 + 111001 (+7 + -7)
( )
( )
1111 −0001 (-1 – 1)
0000 −1010 (0 −−6)
Now we’ll solve them all using the processes mentioned before (We won’t be aligning them for simplicities sake)
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19. Exercise 1 Solutions 1110 + 1111 000111 + 111001 1111 −0001 0000 −1010
Step 1: Left pad any numbers that need to be
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20. Exercise 1 Solutions 1110 + 1111 000111 + 111001 1111+1111 0000+ 0110
Step 2: Convert subtractions to additions
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21. Exercise 1 Solutions 1110+1111 Carry: 0, Remainder: 1
Step 3: Add from right to left
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22. Exercise 1 Solutions 1110+1111 Carry: 10, Remainder: 01
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23. Exercise 1 Solutions 1110+1111 Carry: 110, Remainder: 101
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24. Exercise 1 Solutions 1110+1111 Carry: 1110, Remainder: 1101
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25. Exercise 1 Solutions 1110+1111 Carry: 1110, Remainder: (1)1101
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26. Exercise 1 Solutions 1110+1111 Carry: 1110, Remainder: (1)1101
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27. Exercise 1 Solutions 1110+1111 Carry: 1110, Remainder: (1)1101
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28. Exercise 1 Solutions 1110+1111 = 1101 000111 + 111001 = 000000
= 1101 =
= 1110
= 0110
Step 4: Truncate carry bits
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29. Exercise 1 Solutions 1110+1111 = 1101 000111 + 111001 = 000000
= 1101 =
1111−0001 = 1110
0000−1010 = 0110
Step 5: Revert back to original equations
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30. Exercise 1 Solutions −2+−1 = -3 +7+−7 = 0 −1−1 = -2 0−−6 = 6
−2+−1 = -3
+7+−7 = 0
−1−1 = -2
0−−6 = 6
Step 6: Convert to decimal to check answer
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31. Carry and Overflow Carry Flag: overflow flag:
Set in the following:
= (1)0000 (MSB carried out)
0000 – 0001 = (MSB carried in)
overflow flag:
= 1000 (4 + 4 = -8)
= 0000 ( = 0)
In unsigned arithmetic we care about the carry flag
In signed arithmetic we care about the overflow flag
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32. Bit Shifting Left Shift (<<) Right Shift (>>)
Equivalent to multiplying the value by 2 𝑛
(+23) << 1 (n) = (+46)
Right Shift (>>)
Equivalent to dividing, and rounding down, the value by 2 𝑛
(-105) >> 1 (n) = (-53)
This can change depending on if the value being shifted is signed or unsigned
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